Graphics for the Sierpinski Carpet were posted earlier, up to (and including) 5 iterations. I really wanted to get a sixth iteration so I let the code run and run. All I got were warnings from my computer about a script which was either unresponsive or busy. After about 80 minutes I gave up. But that sort of behavior was similar to what I encountered with Sierpinski's Carpet code. That problem was lessened when I  experimented with multiple Graphics() assignments which led to about double the iterations; maybe something similar could be done with Sierpinski Carpet code? It sure seemed like at least 1 more iteration seemed possible.

With a little extra tinkering I was able to get that extra iteration by creating other Graphics() objects that get used when the length gets smaller. The basic idea which I learned from Sierpinski's triangle is that it seems like a "fresh" Graphics object is needed to get the extra iterations. This was worked into the code by making it use the other Graphics() objects when the length got small. The sixth iteration of Sierpinsksi's Carpet has added to the others. I quit trying to get a 7th iteration after 30 minutes.

I've posted both codes for Sierpinski's Carpet on the Python/Sage page so you can compare them. Maybe this will be helpful in things you're working on.

Depending on the code and the iterations, be prepared to wait minutes for your output, so you might want to increase that slider one iteration at a time and see how your computer handles it. The screenshot from the original code is posted here.

I might have gotten a glimpse of 7 iterations (count the squares) but, unfortunately, it wasn't saved to PDF, just PNG. After looking at it, it seems like the squares blend together. Here is the output. Click on the image and then (again) use the magnifying glass.

As promised, a modified version of the code which produced the images of Sierpinkski's Triangle (which were posted on the Graphics page) has been posted on the Python/Sage page. Although my version went up to 9 iterations, I've changed the code so that the maximum iterations is at 4. Not knowing how fast your computer is I figure it is better that you figure out how many iterations are best. Be prepared to wait minutes for the output. Nothing is wrong, it just takes time.

I'm a lousy programmer, but let me warn you about the code. The 3 lines

T1 = Graphics()
T2 = Graphics()
T3 = Graphics()

make the code appear rather clunky. In fact, my first version through didn't have these lines (and the line S=T1+T2+T3), it just used S to handle everything. That caused problems which I don't have the expertise to explain. At 4 iterations my quad core computer with 8 GB of RAM was brought to a standstill; the memory was all used up. Creating T1, T2, and T3 was my attempt to fix the problem. I don't know why it gives me so many additional iterations but modify it at your own risk.

Finally, even in the version which allowed me get 9 iterations, it took minutes to get the output. So experiment slowly with your machine!

The basic idea behind recursion is to identify the process that takes you from one step to the other; this is revealed in the first step. For Sierpinski's triangle you take a black triangle, identify the midpoint of each side, connect the midpoints to create a triangle which you color white. Those midpoints are vertices of the 3 resulting black triangles which are needed when you repeat the process. The essential information in all of this are the 3 starting vertices. From them we can compute the 3 midpoints by the Midpoint Formula: The midpoint of and  is .

The first IF statement: [ if (N>0): ] draws the initial triangle. If there are still more iterations to be done [ if (N>1): ] then there are 3 triangles to create which will need to be processed again, one triangle for each of the 3 corners (A, B, and C of the original triangle).

You can see the results on the Sage Output page.

I stumbled onto Wolfram Demonstrations Project a month or so ago. They've got a lot of demonstrations that have been put together showing the power of Mathematica. Since Mathematica is expensive to buy it seems like they're trying to promote themselves to organizations rather than people. Some of those demonstrations involve high school mathematics, but I found very few that I could imagine an actual high school math teacher would want to use.

One demonstration I liked was Table of Primes, so I created a prime table of my own using Sage Interact. I wanted my "Primes Table" to have several properties:

• Be able to vary the number of rows and columns
• have the text size adjust to the size of the boxes
• have the numbers in the boxes start in the upper lefthand corner of the page
• be big enough so that I could show a "large" number of primes

I've posted a text file with the code on the Python/Sage page; copy it into a Sage cell and press Shift and Enter at the same time to get this (assuming you set the rows to 9 and columns to 35):

Let's look at some areas of the code:

def the_primes(rows = (9..55), columns = (9..55)):

creates 2 sliders, the row slider will go from 9 through 55 inclusive and the column slider will also go from 9 through 55. Since 55*55=3025, you can study the numbers up to 3025, which is good enough for me. The line

G+=line([(2*i,2*j),(2*(i+1),2*j),(2*(i+1),2*(j+1)),(2*i,2*(j+1))], rgbcolor=(0,0,0))

creates the boxes holding the integers while the complicated looking

a=str(abs((j-1)-rows+1)*columns+i)

creates the proper integer to put into each box so that the smallest number will be in the upper lefthand corner. The sequence of IF statements determines the appropriate size of the text depending on how big the boxes are while

if is_prime(abs((j-1)-rows+1)*columns+i):
G+=polygon([(2*i,2*j),(2*(i+1),2*j),(2*(i+1),2*(j+1)),(2*i,2*(j+1))], rgbcolor=(0,.7,0))

colors the box green if the number inside is prime. Finally, we use "figsize=19" to make the  display big and "aspect_ratio=1" to make sure the squares look like squares, rather than rectangles:

G.show(aspect_ratio=1, figsize=19,axes=False)

With Sage and a little code,  you can create your own demonstrations for free. Now that you understand parts of the code, you can modify it to suit the needs of your own classroom.